Question

Let N be the greatest number that will divide 1305, 4665, 6905, leaving the same remainder in each case. Then the sum of the digits in N is.
(a)4
(b)5
(c)6
(d)8

Answer 1

Hint: Think of the HCF, as whenever the greatest divisor is asked, HCF is required to solve the question.

Complete step-by-step answer:
Given;
N be the greatest number that will divide 1305, 4665, 6905, leaving the same remainder in each case.
Now, we let the remainder to be r.
According to Euclid’s Division lemma, if there are two integers s and t, there is always q and r for which the equation s=tq+r, such that r is positive and less than q.
So, the numbers can be written as;
$1305=Nx+r............(i)$
$4665=Ny+r.............(ii)$
$6905=Nz+r.............(iii)$
Where x, y, z are integers.
Now,
Equation (ii) – Equation (i) :
$4665-1305=Ny-Nx+r-r$
$\begin{align}
  & 3360=N\left( y-x \right) \\
 & \Rightarrow 3360=Nk...........(iv) \\
\end{align}$
Similarly,
Equation (iii) – Equation (ii) :
$\begin{align}
  & 2240=N\left( z-y \right) \\
 & \Rightarrow 2240=Nl...........(v) \\
\end{align}$
Also,
Equation (iii) – Equation (i) :
$\begin{align}
  & 5600=N\left( z-x \right) \\
 & \Rightarrow 5600=Nm...........(vi) \\
\end{align}$
So from equation (iv), (v) and (vi);
We can say that 3360, 2240 and 5600 have N as one of their common factors.
We know, product of all common factors is the HCF.
So, for finding N we first have to find the value of HCF of 3360, 2240 and 5600 and if the HCF is smaller than the smallest number initially mentioned in the question i.e. 1305 then it will be our N.
Using method of factorisation for finding HCF;
\[3360=2\times 2\times 2\times 2\times 2\times 3\times 7\times 5\]
\[2240=2\times 2\times 2\times 2\times 2\times 2\times 7\times 5\]
\[5600=2\times 2\times 2\times 2\times 2\times 5\times 7\times 5\]
\[HCF=2\times 2\times 2\times 2\times 2\times 7\times 5=1120\]
Also, 1120 is smaller than 1305.
$\therefore N=1120$
Now, the sum of the digits of N=1+1+2+0=4.
Hence, the answer is option (a) 4.

Note: If we get HCF of 3360, 2240 and 5600 greater than 1305 then we should have to divide the HCF one by one with all the common factors of 3360, 2240 and 5600 and note the integer we get on dividing the HCF with each common factor. Now from the list of integers select the one which is greatest among the ones which are less than 1305 and these integers would be our N.